Morrey-type Banach spaces, maximal operator and Fourier multipliers

Let $({\mathcal F},\norm{\cdot}_{{\mathcal F}})$ be a Banach space of complex-valued measurable functions on $\Rnn$. In this paper, we consider the Morrey-type Banach space ${\mathcal M}_ {\mathcal F}(p,\lambda)$ as well as its weak type ${\mathcal M}_{\mathcal F}^*(1,\lambda)$. We develop the theory of maximal operator and Fourier multipliers on these spaces.